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The upper critical field in the extended van-Hove scenario
Physica B 294}295 (2001) 492}495
The upper critical "eld in the extended van-Hove scenario R.G. Dias Departamento de Fn& sica, C.Z.C.M. Universidade de Aveiro, 3810-193, Aveiro, Portugal
Abstract We present a study of the superconducting pairing susceptibility K (r) for a two-dimensional isotropic system with 2 a strong power-law divergence in the density of states N()&\> @, b'1. We show that the pair propagator has the scaling form, K (r)"r@\F(¹ @r). An anomalous short-range behavior is found, straightforwardly leading to positive 2 curvature in the upper critical "eld, for b:2 and to a zero temperature divergence, H &¹\> @, for b'2. 2001 Elsevier Science B.V. All rights reserved. PACS: 74.60.Ec Keywords: Extended 2D van-Hove singularity; Two-dimensional BCS superconductor; Superconducting upper critical "eld
1. Introduction One of the most surprising properties of copper oxide superconductors is the upper critical "eld, which has been obtained in magnetoresistance experiments down to very low temperatures in the case of overdoped Tl Ba CuO [1] and >B underdoped YBa Cu O [2]. A very unusual \B H (¹) curve is observed, with very strong positive curvature and no evidence of saturation at low temperatures. This behavior contrasts strongly with the weak coupling BCS result [3] which predicts an approximately parabolic shape for the H curve. Recently, Abrikosov has proposed [4] that these anomalous H curves re#ect a dimensional cross over to quasi-one dimensional superconductivity due to the presence of #at regions in the energy dispersion, that is, extended saddle points. These points have been directly observed in photoemission experiments in copper oxide superconductors [5]. In Abrikosov's approach, the two extended
saddle points in the energy dispersion are replaced by two one-dimensional linear energy dispersions (q )"v ;q and (q )"v ;q . This model is V V W W equivalent to a system of two transverse chains and in this case, the density of states loses its strong energy dependence completely. Furthermore, it is not surprising that he "nds a dimensional crossover in H . In this paper, we argue that these curves do not re#ect a dimensional crossover, but a strong energy dependence of the density of states which results from the presence of these extended saddle points. In the following, we present a study of the superconducting pairing susceptibility for an isotropic two-dimensional system with a strong power-law divergence in the density of states. In the vicinity of the superconducting transition curve, the gap parameter, (r), is small and a perturbation expansion in powers of leads to the semi-classical linearized gap equation [3]
(r)"g drK (r!r)e ArrY\r(r), @
0921-4526/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 7 0 8 - 0
(1)
R.G. Dias / Physica B 294}295 (2001) 492}495
where K (r) is the fermion pair propagator in real @ space for a given temperature ¹"1/, in the absence of the external "eld and the pairing interaction g and is de"ned as K (r, r)" @ 1/ G (r, r)G (r, r), where the Matsubara S \S S Green's function G describes the normal state in S the absence of magnetic "eld. Using Kramers} Kronig relations, K (r) can be rewritten as @ 2 (2) K (r)" d tanh(/2)A(r, )B(r,!), @
with A(k, )"Im G0(k, ) and B(k, )" Re G0(k, ), where G0(k,) is the retarded Green's function in the absence of magnetic "eld and pairing potential. A(r, ) and B(r, ) are the respective Fourier transforms.
a @ r 2
K (r)"r@\F @
@ sign()#k ! D 4 a
(3)
(4)
with
2k 1 tanh[(X)@] 1 F[X]" $ d sin (2) ab @\ 2 @\ # e\S @L L
In a bidimensional system, a van-Hove singularity (VHS) in the density of states results usually from the presence of a saddle point in the energy dispersion (k). In the case of an extended saddle point, (q)&qL !qK, where q"k!k , this leads V W to a power-law divergence in the density of states N()&\> L> K. Such form of the extended saddle point is indicated not only by the direct probing of the energy dispersion using the angle resolved photoemission technique [5], but also by numerical work on the Hubbard model. For instance, Quantum Monte Carlo work by Assaad and Imada [7] in the Hubbard model has found an extended saddle point with a quartic q dependence W at (0,). In order to simplify the problem, we adopt the isotropic dispersion relation: (k)! " a;sign(q)q@, where q"k!k . We also assume that the VHS is pinned at the Fermi level, that is, k "k . The density of states for the above model $ is N()&a\ @b\(! ) @\. Let us assume for now that b is an odd integer. For this simple model, we can compute the spectral function 2k 1 @\ $ ; A(r, )"! r 2ab a ;cos r
and the retarded Greens function G0(r, ), since G0(q, ) is a meromorphic function in the complex q-plane. Note that A(r, )"Im G0(r, ) and B(r, )"Re G0(r, ). After some lengthy but straightforward algebra, one obtains the following expression for the pair propagator:
;sin w 1# cos
2. The pair propagator
493
2 n b
2 # n b
.
When X<1, F[X]&X@\ and for X;1, the function is exponentially small. Note that the thermal length is given by &(a/¹) @. The pair 2 propagator for distances smaller that the thermal length is approximately given by K (r)&r\¹\> @ and therefore, it diverges as 2 the temperature goes to zero. We will show that this will lead to a zero temperature divergence in the upper critical "eld.
3. The upper critical 5eld The zero-"eld critical temperature is easily obtained with the above form for the kernel in the gap equation, ¹ &[k /ga\ @/(b!1)]@ @\. $ The analytical determination of the upper critical curve for the complete temperature range is a di$cult task. So, we obtain the H curves by numer ical solution of the gap equation and study its behavior analytically only at low temperatures or close to ¹ . At zero temperature, the gap equation simpli"es to
(r)&g dr
¹\@ e (rY (& r (&(r), r!r
(5)
where is the magnetic phase acquired by the Cooper pair. The numerical gap solutions fall into
494
R.G. Dias / Physica B 294}295 (2001) 492}495
a universal Gaussian curve, if the x-axis unit is the magnetic length and therefore, with the variable change x "x/(H in the previous equation, the gap function becomes independent of H and we obtain the low temperature scaling of the upper critical "eld, H (¹)&¹\> @ (see Fig. 1). While Eq. (4) for the pair propagator was derived for odd integer b, this equation is qualitatively correct for any value of b*1. For b"1, with the introduction of a cuto!, we recover the usual BCS results and, in particular, H &¹ . For 1)b(2, F[X]&const, if X;1 and therefore, the pair propagator shows a di!erent short range dependence, K (r)&r@\. The pair propagator 2 does not diverge as we decrease the temperature, and with a scaling argument, we can show that now the zero temperature critical "eld is "nite, 1/g&H\@ and thus, H &¹ @. The low-temperature dependence of H can be obtained expanding the pair propagator in powers of ¹, [K (r)!K (r)]/r@\&!(r¹ @) and fol2 lowing Gorkov [3], one obtains H (¹)!H (0) &!¹A @. Curiously, a power-law low-temperature dependence of H has also been suggested by Kotliar and Varma [8] as a consequence of a zero temperature critical point. This dependence, in our picture, results from the scaling form of the pair propagator as given by Eq. (4), but the value of c depends on the speci"c form of the integrand of Eq. (5). One knows that when bP1, the usual expression for the pair propagator should be recovered, which is the one given by Eqs. (4) and (5) only with the sine function in the integrand [9]. For b'2, the exponential term in Eq. (5) dominates and the sine contribution becomes irrelevant. Therefore, when bP1, the low-temperature behavior should be determined by the sine term and as b goes away from 1, the exponential term should take over. With this assumption, c can be determined and the result is c"(2!b)/2, when b&2 and c"(3!b)/2, when bP1. The results obtained until now can be collected into an equation similar to the usual one, [9,10] 1/g" drK (r)e\P& with a qualitative pair suscep@ tibility given by 1 (r¹ @)A K (r)" 2 r\@ sinh[(r¹ @)B]
(6)
Fig. 1. H curves for "3, 5 and 7. Inset: The same curves in a logarithmic scale, showing clearly the low-temperature power-law behavior.
Fig. 2. The qualitative normalized H curves for 1(b(2. As the dispersion relation changes from linear to quadratic, the upper critical curve changes from the usual BCS curve to a curve with strong positive curvature.
with c"0 and d"b!2, if b'2. If 1)b(2, c"d, with c having the behavior described above in order to reproduce the low temperature dependence of the upper critical "eld. In particular, c"1, if b"1 and the usual BCS equation is recovered [9,10]. If bP2, cP0. In Fig. 2, H curves ob tained with this qualitative kernel are displayed. A drastic transformation from conventional
R.G. Dias / Physica B 294}295 (2001) 492}495
495
parabolic-like curves (obtained with c"(3!b)/2) to curves with strong positive curvature (obtained with c"(2!b)/2) is observed as the low temperature exponent 2c/b goes from 2 to 0. In Fig. 2, the experimental H points obtained by Mackenzie et al. for Tl Ba CuO [1] are also >B displayed and "tted with our qualitative H curves. Note that this is a one-parameter "t (c"(2!b)/2), since the normalized curves do not depend on the coupling constant g. A very good agreement is observed for 2c/b"0.45, which according to the picture presented in this paper, implies that the density of states diverges as N()&\ . Most photoemission experiments have found saddle points with quadratic dispersion along one direction and much #atter (higher power dependence) behavior along the other (transversal) direction, indicating therefore a divergence exponent smaller than 1/2. In the case of the saddle point obtained in Ref. [6], a good "t is obtained with a quartic dependence, leading to +0.25 which agrees reasonably with the value extracted from the experimental H curve.
the Fermi level N()&\? on the upper critical "eld of a clean isotropic weak-coupling superconductor. We have shown that for a weak divergence ( less than ), the zero temperature critical "eld is "nite, but strong positive curvature appears in H as approaches . For a stronger divergence ( larger than ), H (¹) has a power-law divergence at ¹"0.
4. Conclusion
[7] [8] [9] [10]
In conclusion, we have studied the e!ect of a power-law divergence of the density of states at
Acknowledgements This research was funded by the Portuguese MCT PRAXIS XXI program under Grant No. 2/2.1/Fis/302/94.
References [1] [2] [3] [4] [5] [6]
A.P Mackenzie et al., Phys. Rev. Lett. 71 (1993) 1238. D.J.C. Walker et al., Phys. Rev. B 51 (1993) 9375. L.P. Gorkov, JEPT 10 (1960) 593. A.A. Abrikosov, Phys. Rev. B 56 (1997) 5112. K. Gofron et al., Phys. Rev. Lett. 73 (1994) 3302. A.A. Abrikosov, J.C. Campuzano, K. Gofron, Physica C 214 (1993) 73. M. Imada, Rev. Mod. Phys. 40 (1998) 1039. G. Kotliar, C.M. Varma, Phys. Rev. Lett. 77 (1996) 2296. R.G. Dias, J.W. Wheatley, Phys. Rev. B 50 (1994) 13 887. R.G. Dias, J.W. Wheatley, Solid State Commun. 98 (1996) 859.
The upper critical "eld in the extended van-Hove scenario R.G. Dias Departamento de Fn& sica, C.Z.C.M. Universidade de Aveiro, 3810-193, Aveiro, Portugal
Abstract We present a study of the superconducting pairing susceptibility K (r) for a two-dimensional isotropic system with 2 a strong power-law divergence in the density of states N()&\> @, b'1. We show that the pair propagator has the scaling form, K (r)"r@\F(¹ @r). An anomalous short-range behavior is found, straightforwardly leading to positive 2 curvature in the upper critical "eld, for b:2 and to a zero temperature divergence, H &¹\> @, for b'2. 2001 Elsevier Science B.V. All rights reserved. PACS: 74.60.Ec Keywords: Extended 2D van-Hove singularity; Two-dimensional BCS superconductor; Superconducting upper critical "eld
1. Introduction One of the most surprising properties of copper oxide superconductors is the upper critical "eld, which has been obtained in magnetoresistance experiments down to very low temperatures in the case of overdoped Tl Ba CuO [1] and >B underdoped YBa Cu O [2]. A very unusual \B H (¹) curve is observed, with very strong positive curvature and no evidence of saturation at low temperatures. This behavior contrasts strongly with the weak coupling BCS result [3] which predicts an approximately parabolic shape for the H curve. Recently, Abrikosov has proposed [4] that these anomalous H curves re#ect a dimensional cross over to quasi-one dimensional superconductivity due to the presence of #at regions in the energy dispersion, that is, extended saddle points. These points have been directly observed in photoemission experiments in copper oxide superconductors [5]. In Abrikosov's approach, the two extended
saddle points in the energy dispersion are replaced by two one-dimensional linear energy dispersions (q )"v ;q and (q )"v ;q . This model is V V W W equivalent to a system of two transverse chains and in this case, the density of states loses its strong energy dependence completely. Furthermore, it is not surprising that he "nds a dimensional crossover in H . In this paper, we argue that these curves do not re#ect a dimensional crossover, but a strong energy dependence of the density of states which results from the presence of these extended saddle points. In the following, we present a study of the superconducting pairing susceptibility for an isotropic two-dimensional system with a strong power-law divergence in the density of states. In the vicinity of the superconducting transition curve, the gap parameter, (r), is small and a perturbation expansion in powers of leads to the semi-classical linearized gap equation [3]
(r)"g drK (r!r)e ArrY\r(r), @
0921-4526/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 7 0 8 - 0
(1)
R.G. Dias / Physica B 294}295 (2001) 492}495
where K (r) is the fermion pair propagator in real @ space for a given temperature ¹"1/, in the absence of the external "eld and the pairing interaction g and is de"ned as K (r, r)" @ 1/ G (r, r)G (r, r), where the Matsubara S \S S Green's function G describes the normal state in S the absence of magnetic "eld. Using Kramers} Kronig relations, K (r) can be rewritten as @ 2 (2) K (r)" d tanh(/2)A(r, )B(r,!), @
with A(k, )"Im G0(k, ) and B(k, )" Re G0(k, ), where G0(k,) is the retarded Green's function in the absence of magnetic "eld and pairing potential. A(r, ) and B(r, ) are the respective Fourier transforms.
a @ r 2
K (r)"r@\F @
@ sign()#k ! D 4 a
(3)
(4)
with
2k 1 tanh[(X)@] 1 F[X]" $ d sin (2) ab @\ 2 @\ # e\S @L L
In a bidimensional system, a van-Hove singularity (VHS) in the density of states results usually from the presence of a saddle point in the energy dispersion (k). In the case of an extended saddle point, (q)&qL !qK, where q"k!k , this leads V W to a power-law divergence in the density of states N()&\> L> K. Such form of the extended saddle point is indicated not only by the direct probing of the energy dispersion using the angle resolved photoemission technique [5], but also by numerical work on the Hubbard model. For instance, Quantum Monte Carlo work by Assaad and Imada [7] in the Hubbard model has found an extended saddle point with a quartic q dependence W at (0,). In order to simplify the problem, we adopt the isotropic dispersion relation: (k)! " a;sign(q)q@, where q"k!k . We also assume that the VHS is pinned at the Fermi level, that is, k "k . The density of states for the above model $ is N()&a\ @b\(! ) @\. Let us assume for now that b is an odd integer. For this simple model, we can compute the spectral function 2k 1 @\ $ ; A(r, )"! r 2ab a ;cos r
and the retarded Greens function G0(r, ), since G0(q, ) is a meromorphic function in the complex q-plane. Note that A(r, )"Im G0(r, ) and B(r, )"Re G0(r, ). After some lengthy but straightforward algebra, one obtains the following expression for the pair propagator:
;sin w 1# cos
2. The pair propagator
493
2 n b
2 # n b
.
When X<1, F[X]&X@\ and for X;1, the function is exponentially small. Note that the thermal length is given by &(a/¹) @. The pair 2 propagator for distances smaller that the thermal length is approximately given by K (r)&r\¹\> @ and therefore, it diverges as 2 the temperature goes to zero. We will show that this will lead to a zero temperature divergence in the upper critical "eld.
3. The upper critical 5eld The zero-"eld critical temperature is easily obtained with the above form for the kernel in the gap equation, ¹ &[k /ga\ @/(b!1)]@ @\. $ The analytical determination of the upper critical curve for the complete temperature range is a di$cult task. So, we obtain the H curves by numer ical solution of the gap equation and study its behavior analytically only at low temperatures or close to ¹ . At zero temperature, the gap equation simpli"es to
(r)&g dr
¹\@ e (rY (& r (&(r), r!r
(5)
where is the magnetic phase acquired by the Cooper pair. The numerical gap solutions fall into
494
R.G. Dias / Physica B 294}295 (2001) 492}495
a universal Gaussian curve, if the x-axis unit is the magnetic length and therefore, with the variable change x "x/(H in the previous equation, the gap function becomes independent of H and we obtain the low temperature scaling of the upper critical "eld, H (¹)&¹\> @ (see Fig. 1). While Eq. (4) for the pair propagator was derived for odd integer b, this equation is qualitatively correct for any value of b*1. For b"1, with the introduction of a cuto!, we recover the usual BCS results and, in particular, H &¹ . For 1)b(2, F[X]&const, if X;1 and therefore, the pair propagator shows a di!erent short range dependence, K (r)&r@\. The pair propagator 2 does not diverge as we decrease the temperature, and with a scaling argument, we can show that now the zero temperature critical "eld is "nite, 1/g&H\@ and thus, H &¹ @. The low-temperature dependence of H can be obtained expanding the pair propagator in powers of ¹, [K (r)!K (r)]/r@\&!(r¹ @) and fol2 lowing Gorkov [3], one obtains H (¹)!H (0) &!¹A @. Curiously, a power-law low-temperature dependence of H has also been suggested by Kotliar and Varma [8] as a consequence of a zero temperature critical point. This dependence, in our picture, results from the scaling form of the pair propagator as given by Eq. (4), but the value of c depends on the speci"c form of the integrand of Eq. (5). One knows that when bP1, the usual expression for the pair propagator should be recovered, which is the one given by Eqs. (4) and (5) only with the sine function in the integrand [9]. For b'2, the exponential term in Eq. (5) dominates and the sine contribution becomes irrelevant. Therefore, when bP1, the low-temperature behavior should be determined by the sine term and as b goes away from 1, the exponential term should take over. With this assumption, c can be determined and the result is c"(2!b)/2, when b&2 and c"(3!b)/2, when bP1. The results obtained until now can be collected into an equation similar to the usual one, [9,10] 1/g" drK (r)e\P& with a qualitative pair suscep@ tibility given by 1 (r¹ @)A K (r)" 2 r\@ sinh[(r¹ @)B]
(6)
Fig. 1. H curves for "3, 5 and 7. Inset: The same curves in a logarithmic scale, showing clearly the low-temperature power-law behavior.
Fig. 2. The qualitative normalized H curves for 1(b(2. As the dispersion relation changes from linear to quadratic, the upper critical curve changes from the usual BCS curve to a curve with strong positive curvature.
with c"0 and d"b!2, if b'2. If 1)b(2, c"d, with c having the behavior described above in order to reproduce the low temperature dependence of the upper critical "eld. In particular, c"1, if b"1 and the usual BCS equation is recovered [9,10]. If bP2, cP0. In Fig. 2, H curves ob tained with this qualitative kernel are displayed. A drastic transformation from conventional
R.G. Dias / Physica B 294}295 (2001) 492}495
495
parabolic-like curves (obtained with c"(3!b)/2) to curves with strong positive curvature (obtained with c"(2!b)/2) is observed as the low temperature exponent 2c/b goes from 2 to 0. In Fig. 2, the experimental H points obtained by Mackenzie et al. for Tl Ba CuO [1] are also >B displayed and "tted with our qualitative H curves. Note that this is a one-parameter "t (c"(2!b)/2), since the normalized curves do not depend on the coupling constant g. A very good agreement is observed for 2c/b"0.45, which according to the picture presented in this paper, implies that the density of states diverges as N()&\ . Most photoemission experiments have found saddle points with quadratic dispersion along one direction and much #atter (higher power dependence) behavior along the other (transversal) direction, indicating therefore a divergence exponent smaller than 1/2. In the case of the saddle point obtained in Ref. [6], a good "t is obtained with a quartic dependence, leading to +0.25 which agrees reasonably with the value extracted from the experimental H curve.
the Fermi level N()&\? on the upper critical "eld of a clean isotropic weak-coupling superconductor. We have shown that for a weak divergence ( less than ), the zero temperature critical "eld is "nite, but strong positive curvature appears in H as approaches . For a stronger divergence ( larger than ), H (¹) has a power-law divergence at ¹"0.
4. Conclusion
[7] [8] [9] [10]
In conclusion, we have studied the e!ect of a power-law divergence of the density of states at
Acknowledgements This research was funded by the Portuguese MCT PRAXIS XXI program under Grant No. 2/2.1/Fis/302/94.
References [1] [2] [3] [4] [5] [6]
A.P Mackenzie et al., Phys. Rev. Lett. 71 (1993) 1238. D.J.C. Walker et al., Phys. Rev. B 51 (1993) 9375. L.P. Gorkov, JEPT 10 (1960) 593. A.A. Abrikosov, Phys. Rev. B 56 (1997) 5112. K. Gofron et al., Phys. Rev. Lett. 73 (1994) 3302. A.A. Abrikosov, J.C. Campuzano, K. Gofron, Physica C 214 (1993) 73. M. Imada, Rev. Mod. Phys. 40 (1998) 1039. G. Kotliar, C.M. Varma, Phys. Rev. Lett. 77 (1996) 2296. R.G. Dias, J.W. Wheatley, Phys. Rev. B 50 (1994) 13 887. R.G. Dias, J.W. Wheatley, Solid State Commun. 98 (1996) 859.